The Frobenius trace kernel is ample on projective spaces and, for curves, surfaces and threefolds, only on Fano varieties of Picard rank 1.
Cartier: Une nouvelle op´ eration sur les formes diff´ erentielles, Comptes rendus de l’Acad´ emie des sciences, Paris 244 (1957), 426–428
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Varieties with ample Frobenius-trace kernel
The Frobenius trace kernel is ample on projective spaces and, for curves, surfaces and threefolds, only on Fano varieties of Picard rank 1.